Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{2} + 18 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 4 + \left(11 a + 6\right)\cdot 19 + \left(a + 15\right)\cdot 19^{2} + \left(18 a + 11\right)\cdot 19^{3} + \left(18 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 2\cdot 19 + 13\cdot 19^{2} + 8\cdot 19^{3} + 17\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 a + 10 + \left(18 a + 10\right)\cdot 19 + \left(7 a + 2\right)\cdot 19^{2} + \left(16 a + 18\right)\cdot 19^{3} + \left(12 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 3 + 17\cdot 19 + \left(11 a + 10\right)\cdot 19^{2} + \left(2 a + 7\right)\cdot 19^{3} + \left(6 a + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 6\cdot 19 + 9\cdot 19^{2} + 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 7 + \left(7 a + 14\right)\cdot 19 + \left(17 a + 5\right)\cdot 19^{2} + 9\cdot 19^{3} + 11\cdot 19^{4} +O\left(19^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,3)(4,5)$ |
| $(1,3,5,4,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,6)(2,3)(4,5)$ | $-2$ |
| $15$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $20$ | $3$ | $(1,5,6)(2,3,4)$ | $1$ |
| $30$ | $4$ | $(3,6,5,4)$ | $0$ |
| $24$ | $5$ | $(1,2,5,6,3)$ | $-1$ |
| $20$ | $6$ | $(1,3,5,4,6,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
